Efficient characterization of symmetrically illuminated symmetric 2D gratings
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 G—PHYSICS
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 G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
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Abstract
Description
This application claims the benefit of U.S. application Ser. No. 10/940,243 filed on Sep. 13, 2004 and entitled “System and Method for Efficient Characterization of Diffracting Structures with Incident Plane Parallel to Grating Lines”.
This invention relates to optical characterization of patterned structures.
Manufacturing processes for producing products usually rely on quantitative measurements to provide information required for process control. Such measurements can be made on the final product, and/or on intermediate stages of the product within the manufacturing process, and/or on tools/fixtures used in the manufacturing process. For example, in semiconductor chip fabrication, measurements can be performed on finished chips (i.e., final product), on a wafer patterned with a photoresist (i.e., intermediate stage), or on a mask (i.e., a tool or fixture). Frequently, as in the case of semiconductor chip fabrication, these measurements are performed on structures having small dimensions. Furthermore, it is highly desirable to perform process control measurements quickly and nondestructively, in order to ensure a minimal impact on the process being controlled. Since optical measurements can be performed quickly, tend to be nondestructive, and can be sensitive to small features, various optical process control measurements have been developed.
Optical process control measurements can often be regarded as methods for measuring parameters of a pattern. For example, a pattern can be a periodic onedimensional or twodimensional grating on the surface of a wafer, and the parameters to measure can include feature dimensions, feature spacings and depth of the grating. To measure these parameters, an optical response of the pattern is measured. For example, reflectance as a function of wavelength can be measured. Typically, the optical response will depend on the parameter (or parameters) of interest in a complicated way such that direct parameter extraction from measured data is impractical. Instead, a mathematical model is typically constructed for the pattern, having the parameters of interest as variables. Within the model, a modeled optical response is calculated corresponding to the measured optical response. The parameters of interest are then determined by adjusting the variables to fit the modeled response to the measured response. Various optical process control measurements differ depending on the measured response(s), and on the kind of mathematical model employed.
A commonlyemployed modeling approach for grating diffraction, known as the rigorous coupled wave analysis (RCWA), is described by Moharam et al. in Journal of the Optical Society of America (JOSA), A12, n5, p 10681076, 1995. The RCWA was first introduced by K. Knop in JOSA, v68, p 1206, 1978, and was later greatly improved by Moharam et al. in the abovereferenced article. Some implementations of the RCWA for 1D gratings are described in U.S. Pat. No. 6,590,656, U.S. 6,483,580, U.S. 5,963,329, and U.S. 5,867,276. The RCWA has been extended to 2D gratings e.g., as considered by Han et al., Applied Optics 31(13), pp 23432352, 1992; and by Lalanne in JOSA A 14(7), pp 15921598, 1997. The use of RCWA modeling for characterizing 2D gratings is also considered in U.S. 2004/007173.
Since a grating is periodic, gratingdiffracted optical fields can be expressed as a superposition of space harmonics, each space harmonic having a different spatial period. The RCWA proceeds by including a finite number of space harmonics in the analysis (e.g., M for a 1D grating and MN for a 2D grating). Increasing M or MN increases accuracy, but requires more computation time, while decreasing M or MN decreases computation time, but provides reduced accuracy. The space harmonics each correspond to a diffraction order, so in a typical 2D case where positive diffraction orders 1 through N_{x}, negative diffraction orders −1 through −N_{x}, and zero order diffraction are to be included for the xdirection in a calculation, we have M=2N_{x}+1. Similarly N=2N_{y}+1 if N_{y }positive and negative orders are included for the y direction.
The time required to perform numerical RCWA calculations is dominated by matrix operations having a calculation time on the order of M^{3 }for a 1D grating or (MN)^{3 }for a 2D grating. Accordingly, various special cases have been considered in the literature where calculation time can be reduced compared to a more general case without reducing accuracy.
For example, in the abovereferenced article by Moharam et al., 1D planar diffraction is identified as a special case of 1D conical diffraction. In 1D planar diffraction, the plane of incidence of the light on the grating is perpendicular to the grating lines, while in 1D conical diffraction, the plane of incidence makes an arbitrary angle with respect to the grating lines. Moharam et al. show that a 1D planar diffraction calculation for N orders requires less than half the computation time of a 1D conical diffraction calculation for N orders. Moharam et al. also indicate that for 1D planar diffraction from a symmetric 1D grating, the matrices to be processed take on special forms (i.e., symmetric for lossless gratings and Hermitian for lossy gratings), which can reduce computation time. Thus, a 1D planar diffraction geometry has typically been used for grating characterization based on RCWA calculations.
Another special case for characterization with RCWA calculations which has been considered is normal incident angle illumination of a symmetric 1D grating, e.g., as considered in U.S. Pat. No. 6,898,537. This case is especially simple, since illumination with normal incident angle is a special case of planar diffraction (i.e., the polarization coupling of conical diffraction does not occur), and illumination with normal incident angle on a symmetric grating leads to symmetric positive and negative diffraction orders. Thus N positive orders, N negative orders and the zero order can be accounted for in this case with only M=N+1 space harmonics. To accomplish this, a specialized RCWA assuming normal incident angle with a symmetric grating is derived from the standard RCWA.
However, the approach of U.S. Pat. No. 6,898,537 requires illumination with normal incident angle on the grating, which leads to practical difficulties. For example, in the common case where the response of interest is a zero order reflection, normal incident angle illumination requires separation of the incident light from the zero order reflected light. Providing such separation (e.g., with a beam splitter) requires additional optical element(s), which undesirably increases system complexity.
A symmetryreduced RCWA for a 1D grating illuminated at offnormal incidence such that the plane of incidence is parallel to the grating lines is described by the present inventors in the abovereferenced application Ser. No. 10/940,243. In this method, symmetry is exploited to account for N positive and N negative diffraction orders (and zero order diffraction) with N+1 space harmonics.
Since 2D RCWA calculations tend to be more time consuming than 1D RCWA calculations, methods of reducing calculation time are of special interest for the 2D case. For example, the abovereferenced U.S. 2004/0078173 application considers the use of a library for storing intermediate results for improving efficiency. However, exploiting symmetry to reduce 2D RCWA calculation time does not appear to be considered in the prior art. Thus it would be an advance in the art to provide characterization of 2D gratings with a symmetryreduced RCWA having decreased calculation time compared to a conventional 2D RCWA.
The present invention provides methods and apparatus for optical characterization based on symmetryreduced 2D RCWA calculations. The invention is applicable to gratings having a grating reflection symmetry plane, such that the grating is invariant under reflection in the symmetry plane. In one embodiment of the invention, the sample is illuminated at normal incidence or at a nonzero angle of incidence such that the plane of incidence is parallel to or identical with the symmetry plane. Since this illumination is consistent with the grating symmetry, the various diffracted field components are either symmetric or antisymmetric with respect to the grating symmetry plane. This symmetry is exploited to provide a symmetryreduced 2D RCWA having reduced matrix dimension that is mathematically equivalent to a conventional 2D RCWA. For RCWA calculations including a large number of diffracted orders, the matrix dimension is reduced by about a factor of two.
In another embodiment of the invention, the grating has an additional reflection symmetry plane. For example, let the grating lie in the xy plane and have reflection symmetry in both the xz and yz planes of an orthogonal Cartesian coordinate system. Normal incidence illumination of such a grating leads to diffracted field components which are symmetric or antisymmetric with respect to both symmetry planes. This symmetry is exploited to provide a symmetryreduced 2D RCWA for normal incidence having reduced matrix dimension that is mathematically equivalent to a conventional 2D RCWA. For RCWA calculations including a large number of diffracted orders, the matrix dimension is reduced by about a factor of four for normal incidence.
Other embodiments of the invention relate to use of a symmetryreduced normal incidence RCWA calculation to provide approximate results for a grating having two symmetry planes and illuminated at a nonzero angle of incidence. The accuracy of this approximation can be improved by modifying either the grating depth or the grating refractive index to account for the effect of incident angle on optical path length.
Grating 110 is assumed to be symmetric in a grating reflection plane 130, such that grating 110 is not changed by a geometrical reflection in plane 130. Effectively exploiting this symmetry of the grating is a key aspect of the invention. The light from source 102 is incident on grating 110 at normal incidence or at a nonzero angle of incidence θ. The plane of incidence (i.e., the plane containing the surface normal and the incident light wave vector) is parallel to (or identical with) the grating reflection plane 130. For normal incidence illumination, such alignment of the plane of incidence and the reflection plane is always possible. Equivalently, the incident optical wave vector lies within grating reflection plane 130 in all cases (i.e., both normal incidence and nonnormal incidence). On
It is often convenient to regard grating 110 as separating region I having index n_{I }from region II having index n_{II}. Although the example of
A detector 104 receives radiation diffracted by the grating. A processor 106 receives a measured response from detector 104 and provides a corresponding modeled response. Adjustment of parameters of the modeled response to achieve a good fit between the measured and modeled responses provides estimates of corresponding grating parameters. For example, these model/grating parameters can include feature width, feature length, feature radius, feature area, grating period (in either direction), grating depth (d on
Processor 106, which can include any combination of hardware and/or software, performs a symmetry reduced 2D RCWA calculation. The applicability of a symmetry reduced 2D RCWA calculation is based on grating symmetry (i.e., reflection plane 130) and on symmetric illumination of the 2D grating (i.e., reflection plane 130 is also the plane of incidence). Thus symmetry reduced 2D RCWA methods are another key aspect of the invention. Symmetry reduced RCWA methods for 2D gratings are best appreciated in connection with conventional 2D grating RCWA methods. Accordingly, a mathematical development of both conventional and symmetry reduced 2D RCWA methods is given in Appendix A. More specifically, Equations A1 through A38c relate to conventional 2D RCWA calculations, while Equations A39A54 relate to symmetry reduced RCWA methods in accordance with embodiments of the invention.
For a conventional 2D RCWA including M diffraction orders in x and N diffraction orders in y, the matrix eigenvalue problem (e.g., as in Eq. A23) has matrix dimension 2MN. Thus if M=2N_{x}+1 and N=2N_{y}+1 (i.e., N_{x }positive and negative diffracted orders are included for x and N_{y }positive and negative orders are included for y), the conventional RCWA eigenvalue problem has matrix dimension 2(2N_{x}+1)(2N_{y}+1). In accordance with the invention, a symmetryreduced RCWA can account for the same set of diffraction orders with an eigenvalue problem having matrix dimension 2(2N_{x}+1)(N_{y}+1) for nonnormal incident light (e.g., as in Eqs. A42 and A45). For normal incidence, the matrix dimension can be further reduced, as described later.
This reduction of matrix dimension is best appreciated in connection with
In view of these two possibilities (symmetry or antisymmetry) it is convenient to define matrix reduction rules as follows. A positive reduction rule R^{+} is given by
R _{mn,pq} ^{+} =F _{mn,pq} +F _{mn,p(−q) }for q≠0, and (1a)
R_{mn,p0} ^{+}=F_{mn,p0 }for q=0, (1b)
where −N_{x}≦m,p≦N_{x}, 0≦n,q≦N_{y}. A negative reduction rule R^{−} is given by
R _{mn,pq} ^{−} =F _{mn,pq} −F _{mn,p(−q) }for q≠0, and (2a)
R_{mn,p0} ^{−}=F_{mn,p0 }for q=0. (2b)
In these equations, F is a full matrix having dimension (2N_{x}+1)(2N_{y}+1), and R^{+} (or R^{−}) is a corresponding reduced matrix having dimension (2N_{x}+1)(N_{y}+1). Such reduction in matrix size is a key advantage of the invention, since it significantly reduces RCWA calculation time.
Thus in the spolarized example of
Similarly, for ppolarized incident radiation ycomponent electric field matrices (i.e., R_{y}, T_{y}, and S_{y}) and xcomponent magnetic field matrices (i.e., U_{x}) are reduced according to reduction rule R^{−} and xcomponent electric field matrices (i.e., R_{x}, T_{x}, and S_{x}) and ycomponent magnetic field matrices (i.e., U_{y}) are reduced according to reduction rule R^{+}. In this case as well, matrices for nonfield quantities (e.g., wavenumbers K_{x }and K_{y}, permittivity E, and inverse permittivity E_{inv}) are reduced using rules R^{+} and/or R^{−}. Details of this matrix reduction for ppolarization are given in Eqs. A43A45, where Eq. A45 shows a reduced matrix eigenvalue problem having dimension 2(2N_{x}+1)(N_{y}+1).
Preferably, light from source 102 is either spolarized or ppolarized, so that RCWA matrix reduction can proceed as indicated above. This procedure is generally referred to as preselection of field components. However, the invention can also be practiced with incident light that has an unknown state or degree of polarization. In such cases, the incident light can be modeled as an appropriate combination of s and p polarized light.
Further reduction of RCWA matrix size is possible for gratings which are illuminated at normal incidence and which have reflection symmetry in the yz plane (in addition to the xz reflection symmetry assumed above). In this special case, the x diffraction orders are related to each other by relations similar to those considered above for the y diffraction orders.
For positive reduction in both x and y, rule R^{++} is given by
R _{mn,pq} ^{++} =F _{mn,pq} +F _{mn,p(−q)} +F _{mn,(−p)q} +F _{mn,(−p)(−q)},
R _{mn,p0} ^{++} =F _{mn,p0} +F _{mn,(−p)0},
R _{mn,0q} ^{++} =F _{mn,0q} +F _{mn,0(−q)},
R_{mn,00} ^{++}=F_{mn,00}, (3a)
where −N_{x}≦m≦N_{x}, −N_{y}≦n≦N_{y}, 1≦p≦N_{x}, 1≦q≦N_{y}. For negative reduction in both x and y, rule R^{−−} is given by
R _{mn,pq} ^{−−} =F _{mn,pq} −F _{mn,p(−q)} −F _{mn,(−p)q} +F _{mn,(−p)(−q)},
R _{mn,p0} ^{−−} =F _{mn,p0} −F _{mn,(−p)0},
R _{mn,0q} ^{−−} =F _{mn,0q} −F _{mn,0(−q)},
R_{mn,00} ^{−−}=F_{mn,00}, (3b)
For positive reduction in x and negative reduction in y, rule R^{+−} is given by
R _{mn,pq} ^{+−} =F _{mn,pq} −F _{mn,p(−q)} +F _{mn,(−p)q} −F _{mn,(−p)(−q)},
R _{mn,p0} ^{+−} =F _{mn,p0} +F _{mn,(−p)0},
R _{mn,0q} ^{+−} =F _{mn,0q} −F _{mn,0(−q)},
R_{mn,00} ^{+−}=F_{mn,00}, (3c)
For negative reduction in x and positive reduction in y, rule R^{−+} is given by
R _{mn,pq} ^{−+} =F _{mn,pq} +F _{mn,p(−q)} −F _{mn,(−p)q} −F _{mn,(−p)(−q)},
R _{mn,p0} ^{−+} =F _{mn,p0} −F _{mn,(−p)0},
R _{mn,0q} ^{−+} =F _{mn,0q} +F _{mn,0(−q)},
R_{mn,00} ^{−+}=F_{mn,00}, (3d)
These reduction rules each relate a full matrix F having dimension (2N_{x}+1)(2N_{y}+1) to a corresponding reduced matrix having dimension (N_{x}+1)(N_{y}+1).
Preselection according to input polarization is preferred. Thus for spolarized (i.e., ypolarized) incidence, electric field matrices reduce according to rule R^{−+} and magnetic field matrices reduce according to rule R^{+−}. For ppolarized (i.e., xpolarized) incidence, electric field matrices reduce according to rule R^{+−} and magnetic field matrices reduce according to rule R^{−+}. Other matrix quantities reduce according to rules R^{++}, R^{−−}, R^{+−}, and/or R^{−+} as indicated in Eqs. A46A50 (spolarization) and Eqs. A51A54 (ppolarization).
Normal incidence symmetry reduction of an RCWA can be exploited for characterization in two different ways. The first way is to illuminate the sample under test at normal incidence and employ the corresponding symmetry reduced normal incidence RCWA. The second way is to illuminate the sample under test at a small nonnormal angle of incidence, and use a symmetryreduced normal incidence RCWA to approximately model this physical situation. In favorable cases, the resulting approximation errors are negligible. Since normal incidence illumination can be problematic in practice (e.g. the problem of separating reflected light from transmitted light arises), this second approach is of considerable interest. Optionally, modifications can be made to the grating depth or refractive index in order to improve the accuracy of the normal incidence angle approximation.
The modified d (or revised d) normal incident angle approximation is based on replacing d with d·cos θ in a symmetry reduced normal incidence RCWA. This approach ensures that the change in phase kd due to the offnormal incident angle is accounted for by altering d. For a simple binary grating (e.g., an airSi grating), only the feature depth needs to be revised in this method, since the nonfeature depth in this case has no independent significance. For gratings having multiple regions with different indices n_{i }and the same depth d, an average
is employed, where θ_{i }is defined by n_{i }sin θ_{i}=n_{I }sin θ and f_{i }is the filling factor for region i. Thus in this case, d is replaced with d cos
Instead of revising the depth of the grating, it is also possible to revise its refractive indices to improve the accuracy of the normal incident angle approximation. The revised index approach is based on setting n_{i }to n_{i }cos θ_{i }for all materials (indexed by i) making up the grating structure (included the substrate) in a symmetry reduced normal incidence RCWA. Here the angles θ_{i }are propagation angles in each region as determined by Snell's law n_{i }sin θ_{i}=n_{i }sin θ. Each refractive index is revised to an “effective index” that provides the proper phase shift in the zdirection. The revised index approach is similar to the revised d approach, except that the cos (θ) factors are applied to the indices instead of to the depth.
Further details and examples of these methods for improving the accuracy of normal incidence angle approximations are considered in connection with 1D grating characterization in U.S. patent application Ser. No. 10/940,243 by the present inventors.
The preceding description has focused on RCWA matrix reductions based on symmetry, which is a key aspect of the invention. Since the invention is based on exploiting symmetry, it is applicable in conjunction with various modifications or refinements of the basic 2D RCWA. For example, the invention is applicable to variants of the RCWA developed for multilayer gratings (e.g., as considered in Eqs. A32A34). For the singlelayer or multilayer gratings, RCWA efficiency can be improved by performing a partial calculation of only reflectances instead of a full calculation including both reflectances and transmittances, provided no transmittances are of interest.
The invention can also be employed in connection with the RCWA given by Lalanne to improve computational efficiency. Lalanne's method is based on an RCWA formalism where the relative contributions of the permittivity matrix E and the inverse permittivity matrix E_{inv }to the RCWA are determined by an adjustable parameter α. For 2D rectangular gratings, α has a preferred value given by ƒ_{y}Λ_{y}/(ƒ_{x}Λ_{x}+ƒ_{y}Λ_{y}). The conventional RCWA development given in Eqs. A1A34 includes Lalanne's α parameter, as do the examples of the present invention given in Eqs. A39A54. Use of the Lalanne RCWA formalism is preferred but not required in practicing the invention.
Although exact symmetry is assumed to derive the various reduced RCWAs, the invention is not restricted to characterization of exactly symmetric gratings. Instead, it is sufficient for the grating to be substantially symmetric in the relevant symmetry plane or planes. The level of grating symmetry needed for sufficient accuracy will be applicationdependent, and can be evaluated for particular applications by an art worker. In general terms, the required illumination symmetry for practicing the invention is that the incident optical wave vector be within a reflection symmetry plane of the 2D grating. Both the offnormal incidence and normal incidence cases considered above are special cases of this wave vector geometry.
At first, we consider the diffraction of the 2D grating based on the rigorous RCWA method, which is described for the general case in the paper “Electromagnetic scattering of twodimensional surfacerelief dielectric gratings”, by Soon Ting Han et al. in Applied Optics, vol. 31, no. 13, 23432352, 1992. For 2D gratings, we can always select the incident plane parallel to one of the grating lines. As shown in
{right arrow over (E)} _{inc} ={right arrow over (u)}exp(−j{right arrow over (k)} _{1} ·{right arrow over (r)})={right arrow over (u)}exp[−jk _{0} n _{1}(sin θ·x+cos θ·z)] (A1)
where
{right arrow over (u)}=cos ψ cos θ·{right arrow over (e)} _{x}+sin ψ·{right arrow over (e)} _{y}−cos ψ sin θ·{right arrow over (e)} _{z} (A2)
k_{0}=2π/λ, λ is the wavelength of light in free space, and ψ=0°, 90° corresponds to p (TM mode) and s (TE mode) polarization respectively. In regions I and II (cover and substrate), the normalized electric fields are
Here {right arrow over (R)}_{m,n}, {right arrow over (T)}_{m,n }are the normalized mn order reflection (reflectance) and transmission field (transmittance) in I and II regions of
In the grating region, the x, y components of the fields can be shown as the Fourier expression (z components are not independent)
where
{right arrow over (σ)}_{mn} =k _{xm} {right arrow over (e)} _{x} +k _{yn} {right arrow over (e)} _{y}=(k _{x0} −mK _{x}){right arrow over (e)} _{x}+(k _{y0} −nK _{y}){right arrow over (e)} _{y} (A7)
k_{x0}=k_{0}n_{I }sin θ cos φ=k_{0}n_{I }sin θ, k_{y0}=k_{0}n_{I }sin θ sin φ=0; K_{x}=2π/Λ_{x}, K_{y}=2π/Λ_{y}, and Λ_{x}, Λ_{y }are grating periods in x, y directions respectively. With the boundary conditions at z=0, z=d and by considering the zero order case m=n=0 in region I, we can find
{right arrow over (k)} _{I,mn}={right arrow over (σ)}_{mn} −k _{zI,mn} {right arrow over (e)} _{z} (A8a)
{right arrow over (k)} _{II,mn}={right arrow over (σ)}_{mn} +k _{zII,mn} {right arrow over (e)} _{z} (A8b)
and
In the grating region, the relative permittivity and its inverse function can be written as
For any given function ε(x,y), ε_{mn}, a_{mn }can be found.
For example, a 2Dgrating with rectangular pixels has coefficients given by
Here n_{1}, n_{2 }are refractive index of the two materials within the grating, and f_{x}, f_{y }are the filling factor of n_{2 }in the period Λ_{x}, Λ_{y }separately. The coefficient of its inverse function is:
With the Fourier expression, the Maxwell equations for the field in the grating region can be expressed in matrix form (as in the paper “Highly improved convergence of the coupledwave method for TM polarization” by Philippe Lalanne et al in JOSA A, Vol. 13 Issue 4, p. 779, 1996).
Here matrices E(mn, pq) and E_{inv}(mn, pq) are constructed by the element ε_{mp,nq}, a_{mp,nq }respectively, z′=z/k_{0}, K_{x }is a diagonal MN×MN matrix with elements of K_{x}(mn,mn)=k_{xm}; and K_{y }is a diagonal MN×MN matrix with elements of K_{y}(mn,mn)=k_{yn}; for n=−N_{x}, . . . , 0, . . . , N_{x }(N=2N_{x}+1), m=−N_{y}, . . . , 0, . . . , N_{y }(M=2N_{y}+1).
For improving convergence rate, eq. (A13) can be rewritten (as in the paper “Improved formulation of the coupledwave method for twodimensional gratings” by Philippe Lalanne et al in JOSA A, 14 Issue 7, p. 1592, 1997)
for 2D grating (and include the special case of f_{x}=0, f_{y}=0). For 1D grating or a uniform layer, α can be written as
Let
and then, S_{t }can be expressed as Fourier series with its space harmonics
where 2MN=2·M·N, and M, N are the maximum retained diffractive number in x, y direction separately. Here i=1, . . . , M·N and i=M·N+1, . . . , 2·M·N corresponds to the S_{x}, S_{y }separately. The eigenvalue equation can be written as
A _{B} =B _{1} B _{2} ·W=W·Q (A21)
and
V=B _{2} ·W·q ^{−1} =B _{1} ^{−1} Wq (A22)
where
Here B=K_{x}E^{−1}K_{x}−I, D=K_{y}E^{−1}K_{y}−I, and q is a diagonal matrix with the element q_{m }(q_{m}=square root of the elements Q_{m }in matrix Q).
Consider the fieldamplitude vector and the corresponding wave vector in the cover and substrate region are orthogonal with each other
{right arrow over (k)} _{I,mn} ·{right arrow over (R)} _{mn}=0, {right arrow over (k)} _{II,mn} ·{right arrow over (T)} _{mn}=0 (A24)
which leads to
Combined with the Maxwell equations and the boundary conditions at the interface of the grating with regions I and II, we have
are 2MN×1 matrices, and
are 2MN×2MN matrices. Here X_{yI }ia an MN×MN diagonal matrix with the diagonal elements of k_{xm}k_{yn}/(k_{zI,mn}k_{0}), X_{I }is a similar matrix having diagonal elements of (k_{zI,mn} ^{2}+k_{xm} ^{2})/(k_{zI,mn}k_{0}), Y_{I }is a similar matrix having diagonal elements of (k_{zI,mn} ^{2}+k_{yn} ^{2})/(k_{zI,mn}k_{0}), D_{1E}, D_{1M }are MN×1 matrices with the elements of sin ψδ_{m0}δ_{n0}, cos ψ cos θδ_{m0}δ_{n0 }respectively, and D_{2E}, D_{2M }are MN×1 matrices with the elements of n_{I }cos ψδ_{m0}δ_{n0}, n_{I }cos θ sin ψδ_{m0}δ_{n0}.
At the boundary z=d, let x_{yII }be an MN×MN diagonal matrix having diagonal elements of k_{xm}k_{yn}/(k_{zII,mn}k_{0}), X_{II }be a similar matrix having diagonal elements of (k_{zII,mn} ^{2}+k_{xm} ^{2})/(k_{zII,mn}k_{0}), and Y_{II }be a similar matrix having diagonal elements of (k_{zII,mn} ^{2}+k_{yn} ^{2})/(k_{zII,mn}k_{0}). We can then find the matrix equation
is a 2MN×1 matrix, and where
is a 2MN×2MN matrix. For a known grating structure and illuminated beam, the reflectance and transmittance can be found by solving Eqs. (A26), (A29).
For a multilayer structure, d_{i}(i=1, . . . , L) are the thicknesses of each layer. At the boundary between l−1 and l layers (z=d_{l1}), the boundary condition can be written as the same form as for a 1D grating (described in the paper “Stable implementation of the rigorous coupledwave analysis for surfacerelief gratings: enhanced transmittance matrix approach” by M. G. Moharam et al in J. Opt. Soc. Amer. Vol. 12, no. 5, 1077, 1995), which gives
In this case, Eqs. (A26), (A29) are rewritten as
Now, both R_{i}, T_{i }can be found by solving above three equations.
Matrix of Dielectric of the 2D Grating with the Circle Media Interface
As shown in
Since the coefficient of the inverse function of relative permittivity is similarly determined, we have
Reduced Order in y
As shown in
In the case of 1D grating with normal incident beam, all the field component are symmetric (as shown in Nanometrics patent U.S. Pat. No. 6,898,537B1). The problem for nonnormal incidence on a 2D grating is different. By tracing the rays of the diffraction within the grating layer (as in
For spolarization (ψ=90°), in which the electric field of the incident beam is perpendicular with the incident plane (as in
S_{y,m(−n)}=S_{y,mn}, U_{x,m(−n)}=U_{x,mn}
R_{y,m(−n)}=R_{y,mn}, T_{y,m(−n)}=T_{y,mn}, K_{y,m(−n)}=K_{y,mn} (A39)
and R_{x}, S_{x}, X_{x}, U_{y}, K_{x }are antisymmetric
S_{x,m(−n)}=−S_{x,mn}, U_{y,m(−n)}=−U_{y,mn}
R_{x,m(−n)}=−R_{x,mn}, T_{x,m(−n)}=−T_{x,mn}, K_{x,m(−n)}=−K_{x,mn} (A40)
Similar to the 1D grating case (as considered in patent application Ser. No. 10/940,243 entitled “System and Method for Efficient Characterization of Diffracting Structures with Incident Plane Parallel to Grating Lines”, filed Sep. 13, 2004 and assigned to Assignee of the present invention), a reduced M(N_{y}+1) order matrix E with the symmetric and antisymmetric field can be constructed by the following rules:
With these transformations, all the related matrices developed from above matrices are all reduced in order from 2MN to 2M(N_{y}+1). The eigen matrix in eq. (A23) can be rewritten as
Here B_{−r}=K_{xr}E_{−r} ^{−1}K_{xr}−I_{r}, D_{−r}=K_{yr}E_{−r} ^{−1}K_{yr}−I_{r}, I_{r }is the M(N_{y}+1) order identity matrix, K_{xr}, K_{yr }are M(N_{y}+1) order diagonal matrices with −N_{y }diffraction orders ignored.
For ppolarization (ψ=0°), R_{x}, T_{x}, S_{x}, U_{y}, K_{y }are symmetric
S_{x,m(−n)}=S_{x,mn}, U_{y,m(−n)}=U_{y,mn}
R_{x,m(−n)}=R_{x,mn}, T_{x,m(−n)}=T_{x,mn}, K_{y,m(−n)}=K_{y,mn} (A43)
and R_{y}, T_{y }S_{y}, U_{x}, K_{x }are antisymmetric
S_{y,m(−n)}=−S_{y,mn}, U_{x,m(−n)}=−U_{x,mn}
R_{y,m(−n)}=−R_{y,mn}, T_{y,m(−n)}=−T_{y,mn}, K_{x,m(−n)}=−K_{x,mn} (A44)
Eq. (A42) can be written as
Here B_{r}=K_{xr}E_{r} ^{−1}K_{xr}−I_{r}, and D_{r}=K_{yr}E_{r} ^{−1}K_{yr}−I_{r}. The differences between Eq. (A45) and Eq. (A42) are only the exchanges of E_{r }and E_{−r}, E_{r,inv }and E_{−r,inv}, B_{r }and B_{−r}, and D_{r }and D_{−r}. The reflectance and transmittance of the diffraction can be found by solving the eigen problem of the reduced matrix A_{B }and its boundary conditions (e.g., as shown in eq. (A33), (A34) for multilayer gratings).
Normal Incidence
As shown in
For spolarization (ψ=90°), in which the electric field of the incident beam is perpendicular with the incident plane (as in
S_{y,(−m)n}=S_{y,mn}, U_{x,(−m)n}=U_{x,mn}
R_{y,(−m)n}=R_{y,mn}, T_{y,(−m)n}=T_{y,mn}, K_{y,(−m)n}=K_{y,mn} (A46)
and R_{x}, T_{x}, S_{x}, U_{y}, K_{x }are antisymmetric
S_{x,(−m)n}=−S_{x,mn}, U_{y,(−m)n}=−U_{y,mn}
R_{x,(−m)n}=−R_{x,mn}, T_{x,(−m)n}=−T_{x,mn}, K_{x,(−m)n}=−K_{x,mn} (A47)
Now the reduced (N_{x}+1)(N_{y}+1) order matrix of E with the symmetric and antisymmetric fields can be reconstructed by the following rules:
Eqs. (A13)(A17) can further be written as
The related eigen matrix, corresponding to eq. (A23), (A42) can be rewritten as
Here B_{x} _{ — } _{y}=K_{xN}E_{x} _{ — } _{y} ^{−1}K_{xN}−I_{N}, D_{x} _{ — } _{y}=K_{yN}E_{x} _{ — } _{y} ^{−1}K_{yN}−I_{N}, and I_{N }is the identity matrix with (N_{x}+1)(N_{y}+1) orders. K_{xN}, K_{yN }are (M_{y}+1)(N_{y}+1) order diagonal matrices with −N_{x}, −N_{y }diffraction orders ignored.
For ppolarization (ψ=0°), R_{x}, T_{x}, S_{x}, U_{y}, K_{y }are symmetric
S_{x,(−m)n}=S_{x,mn}, U_{y,(−m)n}=U_{y,mn}
R_{x,(−m)n}=R_{x,mn}, T_{x,(−m)n}=T_{x,mn}, K_{y,(−m)n}=K_{y,mn} (A51)
and R_{y}, T_{y}, S_{y}, U_{x}, K_{x }are antisymmetric
S_{y,(−m)n}=−S_{y,mn}, U_{x,(−m)n}=−U_{x,mn}
R_{y,(−m)n}=−R_{y,mn}, T_{y,(−m)n}=−T_{y,mn}, K_{x,(−m)n}=−K_{x,mn} (A52)
Eq. (A49) can be written as
The reduced matrices of E_{xy}, E_{x} _{ — } _{y}, E_{ — } _{xy}, E_{y} _{ — } _{x}, E_{xy,inv }E_{ — } _{xy,inv }are shown in Eq. (A48). Similarly, the eigen matrix as in eq. (A45) can be rewritten as the 2(N_{x}+1)(N_{y}+1) form
Here B_{y} _{ — } _{x}=K_{xN}E_{y} _{ — } _{x} ^{−1}K_{xN}−I_{N}, and D_{y} _{ — } _{x}=K_{yN}E_{y} _{ — } _{x} ^{−1}K_{yN}−I_{N}. The reflectance and transmittance of different diffraction orders can be found by solving the eigen problem of the reduced matrix A_{B }and its boundary condition as in Eqs. (A26)˜(A34). For this normal incidence case, all of the matrices in these equations are reduced to order 2(N_{x}+1)(N_{y}+1).
Claims (42)
R _{mn,pq} ^{+−} =F _{mn,pq} −F _{mn,p(−q)} +F _{mn,(−p)q} −F _{mn,(−p)(−q)},
R _{mn,p0} ^{+−} =F _{mn,p0} +F _{mn,(−p)0},
R _{mn,0q} ^{+−} =F _{mn,0q} −F _{mn,0(−q)},
R_{mn,00} ^{+−}=F_{mn,00},
R _{mn,pq} ^{−+} =F _{mn,pq} +F _{mn,p(−q)} −F _{mn,(−p)q} −F _{mn,(−p)(−q)},
R _{mn,p0} ^{−+} =F _{mn,p0} −F _{mn,(−p)0},
R _{mn,0q} ^{−+} =F _{mn,0q} +F _{mn,0(−q)},
R_{mn,00} ^{−+}=F_{mn,00},
R _{mn,pq} ^{+−} =F _{mn,pq} −F _{mn,p(−q)} +F _{mn,(−p)q} −F _{mn,(−p)(−q)},
R _{mn,p0} ^{+−} =F _{mn,p0} +F _{mn,(−p)0},
R _{mn,0q} ^{+−} =F _{mn,0q} −F _{mn,0(−q)},
R_{mn,00} ^{+−}=F_{mn,00},
R _{mn,pq} ^{−+} =F _{mn,pq} +F _{mn,p(−q)} −F _{mn,(−p)q} −F _{mn,(−p)(−q)},
R _{mn,p0} ^{−+} =F _{mn,p0} −F _{mn,(−p)0},
R _{mn,0q} ^{−+} =F _{mn,0q} +F _{mn,0(−q)},
R_{mn,00} ^{−+}=F_{mn,00},
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